Theorem. The square root of 2 is an irrational number, i. e. it cannot be expressed as the ratio $\frac{p}{q}$ of two integers $p$ and $q$.

Proof. Suppose the contrary: that it is possible to write $\sqrt 2 = \frac pq$ for some (positive) integers $p$ and $q$. Suppose also that $\frac{p}{q}$ is already the reduced representation, so that $p$ and $q$ have no common divisors.
Then we can square both sides:

$\left(\frac pq\right)^2 = \frac{p^2}{q^2} = 2$

and multiply by $q^2$:

$p^2 = 2q^2$.

So $p^2$ is the double of a natural number ($q^2$), and therefore even. But this can only be if $p$ itself is even. Thus $p$ can be written as $2k$, for some natural number $k$. Inserting this back yields:

$(2k)^2 = 2q^2$

$\Rightarrow 4k^2 = 2q^2$

$\Rightarrow 2k^2 = q^2.$

Now we see that $q^2$ as well is even (twice the natural number $k^2$). So $q$ can also only be even.
But this means that $\frac pq$ can be reduced, since both $p$ and $q$ have the common divisor 2! This contradicts our assumption. Therefore $\sqrt 2$ cannot be a rational number.

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This means that the set of rational numbers $\mathbb Q$ does not fill the number line, and needs to be expanded into the set of real numbers $\mathbb R$.

Even if you don't remember that particular lesson, I can assure that you have sat through it. This proof is an immovable pillar of any middle or high school curriculum. It appears almost to the letter in every textbook, and transcends cultures and ages. It can be traced back, minus the modern notation, to the Pythagoreans of Ancient Greece. (Cue the drowning myth.)

Problems with the Proof

The proof is lacking, not in correctness, but in its place in math class.

It uses algebra to prove a result of arithmetic: namely, the existence of irrational numbers. Very often, the lessons that follow contain very little algebra, but focus on the number line, number sets, set notation etc. So why would we need algebra in the first place? Algebra is fine and important, but we teachers should recognize when it is not only unnecessary, but actively obfuscating. This is the case here.

It uses algebra in an unusual way: the starting point is an equation, but containing two variables, and the equation is still unsolved by the end. Rather, the first line is the one that looks closest to a solved form, and a third variable is introduced. The nonproficient student is easily baffled by this moonwalking calculation and onslaught of letters. The variables are hard to tell apart, as the various expressions resemble each other a lot. Only by the end, and to whoever is still following, does it become clear that $p$, $q$ and $k$ are strawmen and have no actual values.

The result is very narrow: naturally, one might ask about $\sqrt 3$, $\sqrt 5$, etc. Sometimes their irrationality is given as an exercise. But already for $\sqrt 6$, and in general the square root of composite numbers, the proof scheme breaks down. Students are told, or can guess from their calculators, that "most" square roots of the naturals (and rationals) are, indeed, irrational. But why not prove this, too? It can be done, with zero extra effort, using the right approach.

The proof also demonstrates more than necessary: if the goal is merely to show that irrational numbers exist, there are easier examples, closer to what they look like in practice, such as Champernowne's constant 0.12345678910111213141516... (Of course, this presupposes the knowledge that a rational's decimal representation is either terminating or repeating. A valuable result which can also be demonstrated purely arithmetically.)

There is a better way

All in all, while the proof is a nifty piece of mathematics in a curriculum that is sorely lacking when it comes to proofs and abstract reasoning, we can do better.

The crux of the problem, in my view, is the mingling of two topics (irrationals and square roots) that are better treated separately. One can reason about rational and irrational numbers without even mentioning roots, and within arithmetic alone. This allows for a deeper appreciation of the fractional and decimal representation of numbers, their respective advantages and shortcomings, and the concept of number itself.

The fact that square roots are "usually" irrational can be investigated later, and in a manner that gives a simple and precise condition on the radicand while making no unnecessary use of algebra.

Over the next few posts, I will expand on this alternative approach. Stay tuned!